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Theorem eusv2nf 4154
Description: Two ways to express single-valuedness of a class expression A(x). (Contributed by Mario Carneiro, 18-Nov-2016.)
Hypothesis
Ref Expression
eusv2.1 A V
Assertion
Ref Expression
eusv2nf (∃!yx y = AxA)
Distinct variable groups:   x,y   y,A
Allowed substitution hint:   A(x)

Proof of Theorem eusv2nf
StepHypRef Expression
1 nfeu1 1908 . . . 4 y∃!yx y = A
2 nfe1 1382 . . . . . . 7 xx y = A
32nfeu 1916 . . . . . 6 x∃!yx y = A
4 eusv2.1 . . . . . . . . 9 A V
54isseti 2557 . . . . . . . 8 y y = A
6 19.8a 1479 . . . . . . . . 9 (y = Ax y = A)
76ancri 307 . . . . . . . 8 (y = A → (x y = A y = A))
85, 7eximii 1490 . . . . . . 7 y(x y = A y = A)
9 eupick 1976 . . . . . . 7 ((∃!yx y = A y(x y = A y = A)) → (x y = Ay = A))
108, 9mpan2 401 . . . . . 6 (∃!yx y = A → (x y = Ay = A))
113, 10alrimi 1412 . . . . 5 (∃!yx y = Ax(x y = Ay = A))
12 nf3 1556 . . . . 5 (Ⅎx y = Ax(x y = Ay = A))
1311, 12sylibr 137 . . . 4 (∃!yx y = A → Ⅎx y = A)
141, 13alrimi 1412 . . 3 (∃!yx y = Ayx y = A)
15 dfnfc2 3589 . . . 4 (x A V → (xAyx y = A))
1615, 4mpg 1337 . . 3 (xAyx y = A)
1714, 16sylibr 137 . 2 (∃!yx y = AxA)
18 eusvnfb 4152 . . . 4 (∃!yx y = A ↔ (xA A V))
194, 18mpbiran2 847 . . 3 (∃!yx y = AxA)
20 eusv2i 4153 . . 3 (∃!yx y = A∃!yx y = A)
2119, 20sylbir 125 . 2 (xA∃!yx y = A)
2217, 21impbii 117 1 (∃!yx y = AxA)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   = wceq 1242  wnf 1346  wex 1378   wcel 1390  ∃!weu 1897  wnfc 2162  Vcvv 2551
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-sn 3373  df-pr 3374  df-uni 3572
This theorem is referenced by:  eusv2  4155
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